There is an inclusion map $i: SO_m \to SO_{m+1}$ and this induces a homomorphism
$$i_* : \pi_{m-1}(SO_m) \to \pi_{m-1}(SO_{m+1})$$
Now we can think of $\pi_{m-1}(SO_m)$, via the clutching functions, as the set of $m$-bundles over $S^m$. I happen to have an $m$-bundle over $S^m$ that I like, namely the tangent bundle $T^m$; let's call $[T] \in \pi_{m-1}(SO_m)$ the corresponding element.
Why is $\ker(i_*)$ generated by $[T]$?